How big is an atom?

Everyone knows all matter
consists of atoms. The idea is so familiar that few of us can remember when
we first learned it. It was just always there along with the multiplication
tables and the difference between adjectives and adverbs. So it can be hard
to understand how any real scientific doubts could surround the notion of
atoms as recently as 1900. It is a very helpful exercise to a feel for the
size of an atom.

Here's the exercise. Imagine that I offer to give you one atom of gold for every second that has elapsed in
some time period, say, 100 years. How much gold am I offering you?

Perhaps 100 years is not very long. I'll be more generous.
What about one atom of gold for every second
since the time of Democritus in ancience Greece? Or since the
beginning of recorded history? Or since the age of the dinosaurs?

OK. Let's do this properly. The offer is one atom of gold
for every second that has elapsed since the Big
Bang, the beginning of time. How much gold am I offering you?

Take a little time to fix an answer in your thoughts. It
won't be fun if you don't! Done it? Now scroll
down.

What are you thinking? A teaspoon of gold? A bucketful of
gold? A truckful? One atom for every second since the beginning of time.
Don't underestimate it! Second by second, we
pass through the radiation dominated moments after the big bang, the era in
which primordial matter started to collapse into stars, the time that planets
like our earth formed around them; we pass the earliest stirrings of life on
earth, the earliest moments of intelligent thought here; and on and on. And
with each second of these epochs an atom gets added to the offer.

Here is a gold coin weighing one ounce. How many of
these are on offer?

Not even one of these gold coins in on
offer. At right is a fragment that is 1.5% of the original coin. It
is about the smallest fragment that is easy to show.

The astonishing result is that even this fragment is too big. It is 3,000 times too big!

My offer turns out to be very far from
generous. As many atoms of gold as seconds since the big bang amounts
to a speck of gold weighing 0.14 milligrams and worth 0.21 cents.

So what is the moral? It is that atoms are very small indeed (and perhaps also that universe is
not so old). Atoms are so small that we should think differently about them.
That everything is made of atoms is not as simple an assertion as saying that
the smooth mounds of sand that form the dunes at the beach are really made of
fine granules of broken rock. In the case of the sand, we can convince
ourselves that this is right merely by walking over to the sand, spreading a
pinch of it over the palm of our hand and peering down.There is no comparable
way to check that everything is made of atoms. No simple optical microscope
can magnify enough to make individual atoms visible and none ever will. The
wavelength of light is too long to resolve them. It would be like trying to
feel the shape of one grain of sand while wearing a boxing glove!

For a long time, the idea that everything is made of atoms
seemed like a plausible idea, but not one that had direct relevance to
practical science. That certainly seemed to be the attitude of the physical
chemists of 1900. They had developed very beautiful and very powerful
thermodynamically based methods of understanding the physical properties of
matter. With some effort and immensely complicated computations, Maxwell and
Boltzmann could show that some of these properties could be recovered from
the assumption that matter was made of atoms, even though the atoms were so
unimaginably small that no direct observation of them was possible. But why should a physical chemist adopt methods that could
do only a portion of physical chemistry when the existing theory did it all
and with great beauty?

Of course things were about to change with the work of
Einstein, Perrin and others. For they showed that there are phenomena that
can only be explained by the atomic hypothesis. See "Atoms Entropy Quanta: Einstein's
Statistical Physics of 1905" in this Goodies page. That
changed everything.

So now the experts want to see the
sums. Here they are.

Let us set the value of gold at $450 per troy
ounce. One troy ounce is 31.1g. So that comes to $450/31.1= $14.47 per
gram.

The age of the universe is 1.4x1010 years = 4.418x1017
seconds.

Since the atomic weight of gold is 197 and Avogadro's number is
6.02x1023, 4.418x1017 atoms of gold =
(4.418x1017/6.02x1023)x197grams = .00014458 grams.

This weight of gold is worth $14.47 x .00014458 grams = 0.21 cents.

The fragment shown is 0.015x31.1 grams = 0.4665 grams, which is 3,180 times
bigger.

Copyright John D. Norton, June 17, 2006. First
prepared on a transaltantic flight from Berlin to Pittsburgh, June 17,
2006.
Revised January 23, 2007. Thanks to David Kreller for alerting me to
arithmetic errors in a earlier version.